Large-Signal Analysis of a Parametric Second-Subharmonic Oscillator

Abstract

This theoretical analysis finds the steady-state and transient second-subharmonic oscillations of a parametric oscillator described by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\ddot{q} + 2k\dot{q} + (1 - \delta) q + bq^2 = (3 + \delta) \cos 2z</tex> . Harmonic balance yields expressions, with experimentally observed double-valued properties, for the steady-state subharmonic amplitude in terms of pump amplitude and frequency. Reducing the pump frequency below twice the forced resonant frequency (i.e. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta</tex> < 0) provides larger subharmonic amplitude at the expense of a larger minimum pump amplitude required for oscillation. The transient, analyzed by harmonic balance assuming a slowly varying subharmonic amplitude, is portrayed by trajectories in the in-phase vs quadrature amplitude plane. Those singular points which are closer to this plane's origin than the steady-state points are classified using an index theorem. The following results explain the "hysteresis" effects in the double-valued steady state: The origin is a saddle point (unstable) when it is the only such singularity; it is either a stable node or focus when two other such singularities appear, these being saddle points. Numerical computations of trajectories for parameters giving a saddle-point origin show that although a measure of the 10-90 per cent response time is given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> In 9, ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> being the time constant at the origin, measured in subharmonic cycles), this response time does not increase proportionately to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> as the pump frequency is decreased. The ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b/2k = 2</tex> provides unexcessive overshoot in attaining steady state without severely degrading <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> . Experimental checks of the transien and steady state are presented.